Post on 14-Jul-2020
GFD4
V Zeitlin - GFD
Equatorialdynamics in theRSW modelEquations ofmotion andscalingKelvin wavesYanai wavesRossby andinertia-gravitywaves
Two-layer RSWmodel on theequatorial β-plane
Equatorial wavesin the primitiveequations
Tropical cyclones
Chapter 4: Dynamics in the equatorial region
V. Zeitlin
Course GFD M2 OACOS
GFD4
V Zeitlin - GFD
Equatorialdynamics in theRSW modelEquations ofmotion andscalingKelvin wavesYanai wavesRossby andinertia-gravitywaves
Two-layer RSWmodel on theequatorial β-plane
Equatorial wavesin the primitiveequations
Tropical cyclones
Plan
Equatorial dynamics in the RSW modelEquations of motion and scalingKelvin wavesYanai wavesRossby and inertia-gravity waves
Two-layer RSW model on the equatorial β- plane
Equatorial waves in the primitive equations
Tropical cyclones
GFD4
V Zeitlin - GFD
Equatorialdynamics in theRSW modelEquations ofmotion andscalingKelvin wavesYanai wavesRossby andinertia-gravitywaves
Two-layer RSWmodel on theequatorial β-plane
Equatorial wavesin the primitiveequations
Tropical cyclones
RSW model on the equatorial β- plane
∂tv + v · ∇v + βy z ∧ v + g∇h = 0 . (1)
∂th +∇ · (vh) = 0 , (2)
B.C.: decay at y → ±∞
GFD4
V Zeitlin - GFD
Equatorialdynamics in theRSW modelEquations ofmotion andscalingKelvin wavesYanai wavesRossby andinertia-gravitywaves
Two-layer RSWmodel on theequatorial β-plane
Equatorial wavesin the primitiveequations
Tropical cyclones
Equatorial scalingCharacteristic scales:
I Spatial scale - equatorial deformation radius:
L ∼(√
gHβ
) 12
I Time-scale - T ∼ (βL)−1
I Velocity scale - U ∼ g∆HβL2 , where ∆H - typical deviation
of h with respect to HEquatorial Rossby number:
Ro = ε =UβL2 =
∆HH
. (3)
Non-dimensional equations
∂tv + εv · ∇v + y z ∧ v +∇η = 0 , (4)
∂tη +∇ · v + ε∇ · (vη) = 0 , (5)
GFD4
V Zeitlin - GFD
Equatorialdynamics in theRSW modelEquations ofmotion andscalingKelvin wavesYanai wavesRossby andinertia-gravitywaves
Two-layer RSWmodel on theequatorial β-plane
Equatorial wavesin the primitiveequations
Tropical cyclones
Linearisation
ut − yv + hx = 0, (6)vt + yu + hy = 0, (7)ht + ux + vy = 0. (8)
Change of variables
f =12
(u + h); g =12
(u − h). (9)
Equations (6) - (8) are simplified:
ft + fx +12
(vy − yv) = 0, (10)
gt − gx −12
(vy + yv) = 0, (11)
vt + y(f + g) + (f − g)y = 0. (12)
GFD4
V Zeitlin - GFD
Equatorialdynamics in theRSW modelEquations ofmotion andscalingKelvin wavesYanai wavesRossby andinertia-gravitywaves
Two-layer RSWmodel on theequatorial β-plane
Equatorial wavesin the primitiveequations
Tropical cyclones
Parabolic cylindre (Gauss - Hermite) functions
Functional basis φn(y) en −∞ ≤ y ≤ +∞ with decay b.c.:
φ′′n(y) + (2n + 1− y2)φn(y) = 0, (13)
φn(y) =Hn(y)e−
y22√
2nn!√π, (14)
Hn Hermite polynomials:
H0 = 1, H1 = 2y , H2 = 4y2 − 2, . . . . (15)
φ′n + yφn =√2nφn−1, φ′n − yφn = −
√(2n + 1)φn+1 (16)
GFD4
V Zeitlin - GFD
Equatorialdynamics in theRSW modelEquations ofmotion andscalingKelvin wavesYanai wavesRossby andinertia-gravitywaves
Two-layer RSWmodel on theequatorial β-plane
Equatorial wavesin the primitiveequations
Tropical cyclones
Kelvin waves
Particular solution with v ≡ 0 ⇒:
ft + fx = 0, gt−gx = 0, ⇒ f = F (x− t, y), g = G (x + t, y).(17)
y(f + g) + (f − g)y = 0, ⇒ F ∝ e−y22 , G ∝ e+ y2
2 (18)
C.l. à y ±∞ ⇒ G ≡ 0⇒
u = F0(x − t)e−y22 ; h = F0(x − t)e−
y22 ; v = 0. (19)
GFD4
V Zeitlin - GFD
Equatorialdynamics in theRSW modelEquations ofmotion andscalingKelvin wavesYanai wavesRossby andinertia-gravitywaves
Two-layer RSWmodel on theequatorial β-plane
Equatorial wavesin the primitiveequations
Tropical cyclones
Velocity and pressure distribution in a Kelvin wave
GFD4
V Zeitlin - GFD
Equatorialdynamics in theRSW modelEquations ofmotion andscalingKelvin wavesYanai wavesRossby andinertia-gravitywaves
Two-layer RSWmodel on theequatorial β-plane
Equatorial wavesin the primitiveequations
Tropical cyclones
Yanai waves
Particular solution with g = 0, f 6= 0, v 6= 0. From (6) - (8)we get:
ft + fx +12
(vy − yv) = 0, (20)
vy + yv = 0, (21)vt + yf + fy = 0, (22)
Solution by separation of variables:
v = v0(x , t)φ0(y), f = F1(x , t)φ1(y). (23)
GFD4
V Zeitlin - GFD
Equatorialdynamics in theRSW modelEquations ofmotion andscalingKelvin wavesYanai wavesRossby andinertia-gravitywaves
Two-layer RSWmodel on theequatorial β-plane
Equatorial wavesin the primitiveequations
Tropical cyclones
Equations x − t:
F1t + F1x −1√2v0 = 0, v0t +
√2F1 = 0. (24)
Dispersion relation:
ω =k2±√
k2
4+ 1, (25)
GFD4
V Zeitlin - GFD
Equatorialdynamics in theRSW modelEquations ofmotion andscalingKelvin wavesYanai wavesRossby andinertia-gravitywaves
Two-layer RSWmodel on theequatorial β-plane
Equatorial wavesin the primitiveequations
Tropical cyclones
Velocity and pressure distribution in a Yanai wave;eastward propagation
GFD4
V Zeitlin - GFD
Equatorialdynamics in theRSW modelEquations ofmotion andscalingKelvin wavesYanai wavesRossby andinertia-gravitywaves
Two-layer RSWmodel on theequatorial β-plane
Equatorial wavesin the primitiveequations
Tropical cyclones
Rossby and inertia-gravity waves
General case: elimination of u and h (or f and g) in favourof v :
∂t(∇2v − y2v − ∂ttv
)+ ∂xv = 0. (26)
Expansion of v in a series in φn, v =∑
n vn(x , t)φn(y); ⇒
∂t[∂2
xxvn − (2n + 1)vn − ∂2ttvn]
+ ∂xvn = 0. (27)
Fourier - transformation vn(k , t) =∫
dxe−ikxvn(x , t)
∂3ttt vn + (k2 + 2n + 1)∂t vn − ikvn = 0. (28)
GFD4
V Zeitlin - GFD
Equatorialdynamics in theRSW modelEquations ofmotion andscalingKelvin wavesYanai wavesRossby andinertia-gravitywaves
Two-layer RSWmodel on theequatorial β-plane
Equatorial wavesin the primitiveequations
Tropical cyclones
Hence
vn = vn1(k)e−iωn1 t + vn2(k)e−iωn2 t + vn3(k)e−iωn3 t (29)
where ωnα , α = 1, 2, 3 are 3 roots of the dispersion relation:
ω3nα − (k2 + 2n + 1)ωnα − k = 0. (30)
Lowest eigenvalue of ω - Rossby wave. Two others -inertia-gravity waves.
GFD4
V Zeitlin - GFD
Equatorialdynamics in theRSW modelEquations ofmotion andscalingKelvin wavesYanai wavesRossby andinertia-gravitywaves
Two-layer RSWmodel on theequatorial β-plane
Equatorial wavesin the primitiveequations
Tropical cyclones
Velocity and pressure distribution in a Rossbywave; propagation uniquely westward
GFD4
V Zeitlin - GFD
Equatorialdynamics in theRSW modelEquations ofmotion andscalingKelvin wavesYanai wavesRossby andinertia-gravitywaves
Two-layer RSWmodel on theequatorial β-plane
Equatorial wavesin the primitiveequations
Tropical cyclones
Twin equatorial cyclones - Rossby waves
GFD4
V Zeitlin - GFD
Equatorialdynamics in theRSW modelEquations ofmotion andscalingKelvin wavesYanai wavesRossby andinertia-gravitywaves
Two-layer RSWmodel on theequatorial β-plane
Equatorial wavesin the primitiveequations
Tropical cyclones
Velocity and pressure distribution in aninertie-gravity wave, eastward propagation
GFD4
V Zeitlin - GFD
Equatorialdynamics in theRSW modelEquations ofmotion andscalingKelvin wavesYanai wavesRossby andinertia-gravitywaves
Two-layer RSWmodel on theequatorial β-plane
Equatorial wavesin the primitiveequations
Tropical cyclones
Remarques
I Formal application of the dispersion relation at n = −1:
ω3−(k2−1)ω−k = 0,⇒ (ω−k)(ω2+ωk+1) = 0. (31)
Positive root ω = k : Kelvin wave,I Formal application of the dispersion relation at n = 0:
ω3−(k2+1)ω−k = 0,⇒ (ω+k)(ω2−ωk−1) = 0. (32)
Positive root ω2 − ωk − 1 = 0: Yanai wavesI ωn = − k
2n+1+k2 - excellent approximate formula(precision 2%) for the Rossby branch.
GFD4
V Zeitlin - GFD
Equatorialdynamics in theRSW modelEquations ofmotion andscalingKelvin wavesYanai wavesRossby andinertia-gravitywaves
Two-layer RSWmodel on theequatorial β-plane
Equatorial wavesin the primitiveequations
Tropical cyclones
Dispersion and spectral separation of equatorialwaves
−3 −2 −1 0 1 2 30
1
2
3
Relation de dispersion des ondes equatoriales
k(c/2β)1/2
ω/(2β c)1/2
Gravity
Kelvin
Yanai
Rossby
GFD4
V Zeitlin - GFD
Equatorialdynamics in theRSW modelEquations ofmotion andscalingKelvin wavesYanai wavesRossby andinertia-gravitywaves
Two-layer RSWmodel on theequatorial β-plane
Equatorial wavesin the primitiveequations
Tropical cyclones
Adjustement of geopotential anomaly in teequatorial region
GFD4
V Zeitlin - GFD
Equatorialdynamics in theRSW modelEquations ofmotion andscalingKelvin wavesYanai wavesRossby andinertia-gravitywaves
Two-layer RSWmodel on theequatorial β-plane
Equatorial wavesin the primitiveequations
Tropical cyclones
Exercises
I Obtain formula (26),I Demonstrate that for equatorial Rossby waves
(u, v , η) = (U(y),V (y),H(y)) e i(kx−ωt), (33)
V (y) = φn(y), U(y) = −ikφ′n(y)− ωnyφn(y)
ω2n − k2 ,
H(y) = iωnφ
′n(y)− kyφn(y)
ω2n − k2 . (34)
GFD4
V Zeitlin - GFD
Equatorialdynamics in theRSW modelEquations ofmotion andscalingKelvin wavesYanai wavesRossby andinertia-gravitywaves
Two-layer RSWmodel on theequatorial β-plane
Equatorial wavesin the primitiveequations
Tropical cyclones
Equations of the model
∂tvi + vi · ∇vi + βy z ∧ vi +1ρi∇πi = 0 , i = 1, 2; (35)
∂thi +∇ · (hivi ) = 0 (36)
(ρ2 − ρ1)gη = π2 − π1, h1 + h2 = H. (37)
Simplifying hypotheses:
I ρ1 → ρ2, π2 = π1 + ρ1g ′h1, g ′ = g ρ2−ρ1ρ1
I H1 = H2
GFD4
V Zeitlin - GFD
Equatorialdynamics in theRSW modelEquations ofmotion andscalingKelvin wavesYanai wavesRossby andinertia-gravitywaves
Two-layer RSWmodel on theequatorial β-plane
Equatorial wavesin the primitiveequations
Tropical cyclones
Scaling
I Spatial scale - baroclinic equatorial deformation radius :
L ∼(√
g ′Hβ
) 12
I Time-scale - T ∼ (βL)−1
I Velocity scale - U ∼ g ′∆HβL2
I Pressure scale - Pi ∼ ρiUβL2
Barotropic and baroclinic velocities:
vbt =h1v1 + h2v2
H, vbc = v1 − v2 (38)
Barotropic streamfunction:
h1+h2 = const⇒ ∇·(h1v1+h2v2) = H∇·vbt = 0⇒ vbt = z∧∇ψ
GFD4
V Zeitlin - GFD
Equatorialdynamics in theRSW modelEquations ofmotion andscalingKelvin wavesYanai wavesRossby andinertia-gravitywaves
Two-layer RSWmodel on theequatorial β-plane
Equatorial wavesin the primitiveequations
Tropical cyclones
Non-dimensional equations forψ, vbc ≡ v = (u, v), η
∇2ψt + ψx = O(ε) (39)
vt +∇η + y z× v = O(ε) (40)
ηt +∇ · v = O(ε), (41)
where ε = ∆HH - Rossby number.
GFD4
V Zeitlin - GFD
Equatorialdynamics in theRSW modelEquations ofmotion andscalingKelvin wavesYanai wavesRossby andinertia-gravitywaves
Two-layer RSWmodel on theequatorial β-plane
Equatorial wavesin the primitiveequations
Tropical cyclones
Linearised equations
∇2ψt + ψx = 0,vt +∇h + y z× v = 0,
ht +∇ · v = 0. (42)
GFD4
V Zeitlin - GFD
Equatorialdynamics in theRSW modelEquations ofmotion andscalingKelvin wavesYanai wavesRossby andinertia-gravitywaves
Two-layer RSWmodel on theequatorial β-plane
Equatorial wavesin the primitiveequations
Tropical cyclones
Wave solutionsI "Free" barotropic Rossby waves
ψ0 = Aψe i(θ+ly) + c .c .; θ = kx − ωt, (43)
with dispersion
ω = −k/(k2 + l2), (44)
I "Trapped" baroclinic waves:
(u, v , h) = (iUn,Vn, iHn) Ae iθn + c .c .; θn = kx − ωnt(45)
with dispersion
ω3n− (k2 + 2n + 1)ωn− k = 0; n = −1, 0, 1, 2, ... , (46)
- Kelvin, Yanai, Rossby, Inertia-Gravity
Waveguide transparent for barotropicwaves
Exercise: Derive the equations (39 - 41)
GFD4
V Zeitlin - GFD
Equatorialdynamics in theRSW modelEquations ofmotion andscalingKelvin wavesYanai wavesRossby andinertia-gravitywaves
Two-layer RSWmodel on theequatorial β-plane
Equatorial wavesin the primitiveequations
Tropical cyclones
Equatorial waveguide and planetary waves
Interaction free planetary waves trapped equatorial waves
Incoming free planetary wave
Secondary (radiated) planetary wave
Trapped equatorial wave (Rossby and/or Yanai)
Equatorial waveguide
Ω
Ω
equator
GFD4
V Zeitlin - GFD
Equatorialdynamics in theRSW modelEquations ofmotion andscalingKelvin wavesYanai wavesRossby andinertia-gravitywaves
Two-layer RSWmodel on theequatorial β-plane
Equatorial wavesin the primitiveequations
Tropical cyclones
Linearised primitive equations on the equatorial β-plane
ut − βyv + φx = 0,vt + βyu + φy = 0, (47)
ux + vy −(N(z)−2φzt
)z = 0, (48)
Separation of variables 1(u, v , φ) = (u, v , φ)(y , z)e i(kx−ωt):
− iωu − βy v + ikφ = 0,−iωv + βy u + φy = 0, (49)
iku + vy + iω(N(z)−2φz
)z = 0, (50)
GFD4
V Zeitlin - GFD
Equatorialdynamics in theRSW modelEquations ofmotion andscalingKelvin wavesYanai wavesRossby andinertia-gravitywaves
Two-layer RSWmodel on theequatorial β-plane
Equatorial wavesin the primitiveequations
Tropical cyclones
Elimination of u, φ:
(βkω + ω2k2) v + ω2
[(ω2 − β2y2)( vz
N2(z)
)z− vyy
]= 0.
(51)
Separation of variables 2v = χ(z)ν(y)⇒
I
1χ
(χ′
N2(z)
)′= −κ2 = const, (52)
I
−ν′′
ν+ k2 +
βkω
+ (β2y2 − ω2)κ2 = 0 (53)
GFD4
V Zeitlin - GFD
Equatorialdynamics in theRSW modelEquations ofmotion andscalingKelvin wavesYanai wavesRossby andinertia-gravitywaves
Two-layer RSWmodel on theequatorial β-plane
Equatorial wavesin the primitiveequations
Tropical cyclones
Particular case: N = constκ2N2 = l2 = const, vertical modes -harmonic functions
Renormalisation: y →(
Nβl
) 12 y ,
(Nβl
) 12 - equivalent height ⇒
ν ′′(y) +
(ω2lNβ− y2 − k2N
βl− kNωl
)ν(y) = 0 (54)
Expansion in φn(y) ⇒Dispersion relation :
ω3 −[
(2n + 1)Nβl
+k2N2
l2
]ω − βkN2
l2= 0. (55)
GFD4
V Zeitlin - GFD
Equatorialdynamics in theRSW modelEquations ofmotion andscalingKelvin wavesYanai wavesRossby andinertia-gravitywaves
Two-layer RSWmodel on theequatorial β-plane
Equatorial wavesin the primitiveequations
Tropical cyclones
Equatorial waves in Primitive EquationsI Kelvin waves, n = −1:(
ω − kNl
)[ω
(ω +
Nkl
)+βNl
]= 0, (56)
Positive roots: ω = Nl k .
I Yanai waves, n = 0:(ω +
kNl
)[ω
(ω − Nk
l
)− βN
l
]= 0, (57)
Positive roots: ω = N2l k ±
√N2k2
4l2 + βNl .
I Rossby waves, n > 0, lower branch :
ω ≈ − βk
(2n + 1)βlN + k2
. (58)
I Inertia-gravity waves: n > 0, upper branches.
GFD4
V Zeitlin - GFD
Equatorialdynamics in theRSW modelEquations ofmotion andscalingKelvin wavesYanai wavesRossby andinertia-gravitywaves
Two-layer RSWmodel on theequatorial β-plane
Equatorial wavesin the primitiveequations
Tropical cyclones
Extracting equatorial waves from data
top - vertical section along the equator of the eastwardpropagating meridional velocity anomaly, with averagetropopause. Short black line - section of the lower right plot.bottom left - horizontal section of wind and geopotentialanomaly of E0GW, as predicted by shallow water theory;bottom right - horizontal section at 70 hPa.
GFD4
V Zeitlin - GFD
Equatorialdynamics in theRSW modelEquations ofmotion andscalingKelvin wavesYanai wavesRossby andinertia-gravitywaves
Two-layer RSWmodel on theequatorial β-plane
Equatorial wavesin the primitiveequations
Tropical cyclones
Tropical cyclone
GFD4
V Zeitlin - GFD
Equatorialdynamics in theRSW modelEquations ofmotion andscalingKelvin wavesYanai wavesRossby andinertia-gravitywaves
Two-layer RSWmodel on theequatorial β-plane
Equatorial wavesin the primitiveequations
Tropical cyclones
Hydrodynamic characteristics of TC
I Radius of max wind (RMW): 20− 50kmI Max velocity Vmax : 40− 60m/sI Typical value of f (at 20◦N) : 5 · 10−5
I Relative vorticity ζ: up to 100f , typical Rossbynumbers: 10− 40
I Vertical wind distribution: ≈ barotropicI Barotropic Froude number:
Fr = Vmax√gH0
= Ro/√
Bu = O(10−1)
I Radial wind profile: U-shape in the core, decreasing as1/r in the outer region → ≈ constant vorticity coresurrounded by higher vorticity ring, zero vorticity in theouter region.
GFD4
V Zeitlin - GFD
Equatorialdynamics in theRSW modelEquations ofmotion andscalingKelvin wavesYanai wavesRossby andinertia-gravitywaves
Two-layer RSWmodel on theequatorial β-plane
Equatorial wavesin the primitiveequations
Tropical cyclones
Category 3 cyclone’s profile
0 1 2 3 4 5−0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
v
radius
0.92
0.94
0.96
0.98
1
h
12
3
(a) V, H
0 0.5 1 1.5 2 2.5−20
0
20
40
60
80
100
120
radius
ζ
d1 d2
Wr
(b) ζ
Parameters: ratio of relative vorticities ring/core ζrζc
= 105/7,
Roloc = ζrf = 105, Ro = Vmax/fL = 32, Fr = 0.3.
GFD4
V Zeitlin - GFD
Equatorialdynamics in theRSW modelEquations ofmotion andscalingKelvin wavesYanai wavesRossby andinertia-gravitywaves
Two-layer RSWmodel on theequatorial β-plane
Equatorial wavesin the primitiveequations
Tropical cyclones
Simple RSW modelling, polar coordinates
d~vdt
+(f +
vr
)~ez × ~v + g ~∇h = 0, (59)
∂th +1r
(∂r (rhu) + ∂θ(hv)) = 0 (60)
~v = (u~er , v ~eθ). Lagrangian derivative: ddt = ∂
∂t + u∂r + vr ∂θ.
Non-dimensional axisymmetric vortex solution -cyclo-geostrophic equilibrium:(
Ro−1 +V (r)
r
)V (r) =
λ
Fr2 H ′(r) (61)
GFD4
V Zeitlin - GFD
Equatorialdynamics in theRSW modelEquations ofmotion andscalingKelvin wavesYanai wavesRossby andinertia-gravitywaves
Two-layer RSWmodel on theequatorial β-plane
Equatorial wavesin the primitiveequations
Tropical cyclones
Typical evolution of vorticity of a cyclone -barotropic instability
GFD4
V Zeitlin - GFD
Equatorialdynamics in theRSW modelEquations ofmotion andscalingKelvin wavesYanai wavesRossby andinertia-gravitywaves
Two-layer RSWmodel on theequatorial β-plane
Equatorial wavesin the primitiveequations
Tropical cyclones
Mesovortices as observed in the hurricane Isabel